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    Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation


    Beresnevich, Victor and Dickinson, Detta and Velani, Sanju (2001) Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation. Mathematische Annalen, 321 (2). pp. 253-273. ISSN 0025-5831

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    Abstract

    For each real number α, let E(α) denote the set of real numbers with exact order α. A theorem of Güting states that for α≥2 the Hausdorff dimension of E(α) is equal to 2/α. In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem.

    Item Type: Article
    Additional Information: Cite as: Beresnevich, V., Dickinson, D. & Velani, S. Math Ann (2001) 321: 253. https://doi.org/10.1007/s002080100225
    Keywords: Real Number; Linear Form; Critical Exponent; Hausdorff Dimension Hausdorff Measure;
    Academic Unit: Faculty of Science and Engineering > Mathematics and Statistics
    Item ID: 10117
    Identification Number: https://doi.org/10.1007/s002080100225
    Depositing User: Dr. Detta Dickinson
    Date Deposited: 18 Oct 2018 14:08
    Journal or Publication Title: Mathematische Annalen
    Publisher: Springer Verlag
    Refereed: Yes
    URI:

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